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Algebraic solutions of differential equations (p-curvature and the Hodge filtration)

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References

  1. Atiyah, M., Hodge, W.: Integrals of the second kind on an algebraic variety. Annals of Math.62, 56–91 (1955).

    Google Scholar 

  2. Bateman Manuscript Project (Erdelyi Ed.): Higher transcendental functions, vol. 1. New-York: McGraw-Hill 1953.

    Google Scholar 

  3. Cartier, P.: Une nouvelle opération sur les formes différentielles. C. R. Acad. Sci. Paris244, 426–428 (1957).

    Google Scholar 

  4. Cartier, P.: Questions de rationalité des diviseurs en géométrie algébrique. Bull. Soc. Math. France86, 177–251 (1958).

    Google Scholar 

  5. Clemens, H.: Picard-Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities. Trans. Amer. Math. Soc.136, 93–108 (1969).

    Google Scholar 

  6. Deligne, P.: Théorème de Lefschetz et critères de dégénérescence de suites spectrales. Publ. Math. I. H. E. S.35, 107–126 (1969).

    Google Scholar 

  7. Deligne, P.: Cohomologie des intersections complètes. Exposé XI, SGA 7, 1969. Multigraph available from I. H. E. S, 91-Bures-sur-Y vette, France.

  8. Deligne, P.: Travaux de Griffiths. Exposé 376, Séminaire N. Bourbaki, 1969/1970. Lectures Notes in Mathematics180. Berlin-Heidelberg-New York: Springer 1971.

    Google Scholar 

  9. Deligne, P.: Equations différentielles à points singuliers réguliers. Lecture Notes in Mathematics163. Berlin-Heidelberg-New York: Springer 1970.

    Google Scholar 

  10. Deligne, P.: Théorie de Hodge, Publ. Math. I. H. E. S.40 (1971).

  11. Dwork, B.: P-adic cycles. Publ. Math. I. H. E. S.37, 27–115 (1970).

    Google Scholar 

  12. Goursat, E.: L'Equation d'Euler et de Gauss. Paris: Hermann 1936.

    Google Scholar 

  13. Goursat, E.: Intégrales Algébriques. Paris: Hermann 1938.

    Google Scholar 

  14. Griffiths, P.: Periods of integrals on algebraic manifolds: summary of main results and discussion of open problems. Bull. Amer. Math. Soc.75, 2, 228–296 (1970).

    Google Scholar 

  15. Grothendieck, A., Dieudonne, J.: Eléments de géométrie algébrique, Publ. Math. I. H. E. S.4, 8, 11, 17, 20, 24, 28, 32.

  16. Grothendieck, A.: Fondements de la géométrie algébrique. Secrétariat mathématique, 11, rue Pierre Curie, Paris 5o, 1962.

  17. Grothendieck, A.: On the de Rham cohomology of algebraic varieties. Publ. Math. I. H. E. S.29 (1966).

  18. Hartshorne, R.: Residues and duality. Lecture Notes in Mathematics20. Berlin-Heidelberg-New York: Springer 1966.

    Google Scholar 

  19. Hasse, H.: Existenz separabler zyklischer unverzweigter Erweiterungskörper vom Primzahlgradep über elliptischen Funktionenkörpern der Charakteristikp. J. Reine angew. Math.172, 77–85 (1934).

    Google Scholar 

  20. Hasse, H., Witt, E.: Zyklische unverzweigte Erweiterungskörper vom Primzahlgradep über einem algebraischen Funktionenkörper der Charakteristikp. Monatsh. für Math. u. Phys.43, 477–492 (1936).

    Google Scholar 

  21. Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero I, II. Annals of Math.79, 109–326 (1964).

    Google Scholar 

  22. Hironaka, H.: Bimeromorphic smoothing of a complex-analytic space. Preprint available from Mathematics Institute, University of Warwick (1971).

  23. Hochschild, G.: Simple algebras with purely inseparable splitting fields of exponent one. Trans. Amer. Math. Soc.79, 477–489 (1955).

    Google Scholar 

  24. Honda, T.: Differential equations and formal groups. (Preprint). Presented at the 1971 U.S.-Japan Number Theory conference.

  25. Igusa, J.: Class number of a definite quaternion with prime discriminant. Proc. Nat'l. Acad. Sci.44, 312–314 (1958).

    Google Scholar 

  26. Ihara, Y.: Schwarzian equations 1. Preprint, 1971.

  27. Katz, N.: On the differential equations satisfied by period matrices. Publ. Math. I. H. E. S.35 (1968).

  28. Katz, N.: Nilpotent connections and the monodromy theorem: application of a result of Turrittin. Publ. Math. I. H. E. S.39, 355–232 (1970).

    Google Scholar 

  29. Katz, N.: Une formule de congruence pour la fonction ξ. Exposé XXII, SGA 7, 1969. multigraph available from I. H. E. S. 91-Bures-sur-Y vette, France.

  30. Kodaira, K., Spencer, D. C.: On deformations of complex structures, I, II, Annals of Math.67, 328–466 (1958).

    Google Scholar 

  31. Manin, Ju.: Algebraic curves over fields with differentiation. AMS Translations (2),37, 59–78 (1964).

    Google Scholar 

  32. Manin, Ju.: The Hasse-Witt matrix of an algebraic curve. AMS Translations (2)45, 245–264 (1965).

    Google Scholar 

  33. Manin, Ju.: Rational points of algebraic curves over function fields. AMS Translations (2),50, 189–234 (1966).

    Google Scholar 

  34. Mazur, B.: Frobenius and the Hodge filtration, Preprint available form Mathematics Institute, University of Warwick (1971).

  35. Mazur, B.: Frobenius and the Hodge filtration. (To appear.)

  36. Messing, W.: On the nilpotence of the hypergeometric equation. (To appear.)

  37. Messing, W.: The crystals associated to Barsoth-Tate groups; with applications to abelian schemes. Thesis, Princeton, 1971. (To appear.)

  38. Messing, W., Mazur, B.: Cristalline cohomology and the universal extension of an abelian scheme. (To appear.)

  39. Mumford, D.: Abelian varieties. Bombay: Oxford University Press 1971.

    Google Scholar 

  40. Oda, T., Katz, N.: On the differentiation of de Rham cohomology classes with respect to parameters. J. Math. Kyoto Univer.8, 199–213 (1968).

    Google Scholar 

  41. Raynaud, M.: Geométrie algébrique et géométrie analytique. Exposé XII, SGA 1. Lecture Notes in Mathematics224. Berlin-Heidelberg-New York: Springer 1971.

    Google Scholar 

  42. Serre, J.P.: Représentations linéaires des groupes finis. Paris: Hermann 1967.

    Google Scholar 

  43. Serre, J.P.: Sur la topologie des variétés algébriques en caractéristiquep. Symposio International de Topologia Algebraica. Mexico, 1958.

  44. Serre, J.P.: Geométrie algébrique et géométrie analytique. Ann Inst. Fourier. Grenoble6, 1–42 (1956).

    Google Scholar 

  45. Tate, J.:W-C groups overp-adic fields. Exposé 156. Séminaire BOurbaki 1957/1958. New-York: W. A. Benjamin 1966.

    Google Scholar 

  46. Tessier, B., Lejeune, M.: Quelques calculs utiles pour la résolution des singularités. Multigraph available from Centre de Mathématiques, Ecole Polytechnique, 17, rue Descartes, Paris 5o, (1971).

  47. Verdier, J. L.: Base change for twisted inverse image of coherent sheaves. Algebraic geometry, Bombay: Oxford University Press 1968.

    Google Scholar 

  48. Weil, A.: Variétés Kählériennes. Act. Sci. et Ind. 1267, Paris: Hermann 1968.

    Google Scholar 

  49. Weil, A.: Jacobi sums as “Größencharaktere”. Trans. Amer. Math. Soc.73, 487–495 (1952).

    Google Scholar 

  50. Whittaker, E. T., Watson, G.N.: A course of modern analysis. Cambridge: Cambridge University Press 1962.

    Google Scholar 

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Katz, N.M. Algebraic solutions of differential equations (p-curvature and the Hodge filtration). Invent Math 18, 1–118 (1972). https://doi.org/10.1007/BF01389714

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